Major Techniques of Supervised Learning: From Linear Regression to Support Vector Machines

1. Background introduction

Supervised learning is an important branch of machine learning. Its main goal is to use labeled data for model training in order to predict unknown data. In this article, we’ll dive into the main techniques of supervised learning, from linear regression to support vector machines.

1.1 Basic concepts of supervised learning

The basic concepts of supervised learning include training sets, test sets, features, labels, loss functions, etc.

• Training set: A data set consisting of training data used to train the model.
• Test set: A data set consisting of test data used to evaluate the generalization ability of the model.
• Features: Each attribute in the input data is used to describe the data.
• Label: Each attribute in the output data is used to train the model.
• Loss function: used to measure the difference between the model prediction and the actual label, usually a non-negative value, a small value indicates a better prediction, and a large value indicates a poor prediction.

1.2 Main techniques of supervised learning

The main techniques of supervised learning include linear regression, logistic regression, decision trees, random forests, support vector machines, etc.

1.2.1 Linear regression

Linear regression is a simple supervised learning algorithm used to predict continuous target variables. Its core idea is to make the model's prediction of the training data as close as possible to the actual label by finding the optimal weight vector.

1.2.1.1 Core concepts and connections

The core concepts of linear regression include training data, weight vectors, loss functions, etc.

• Training data: A data set consisting of input features and corresponding labels.
• Weight vector: Parameters used to map input features to target variables.
• Loss function: used to measure the difference between model predictions and actual labels. Commonly used loss functions include mean square error (MSE) and root mean square error (RMSE).

1.2.1.2 Detailed explanation of core algorithm principles and specific operation steps as well as mathematical model formulas

The algorithm principle of linear regression is to gradually update the weight vector through the gradient descent method to minimize the loss function. The specific steps are as follows:

1. Initialize the weight vector to a random value.
2. For each training data, the predicted value is calculated.
3. Calculate the difference between the predicted value and the actual label.
4. Update the weight vector to minimize the loss function.
5. Repeat steps 2-4 until convergence.

The mathematical model formula is:

$$and={In}^T\ast x+b$$

$$L=\frac{1}{2n}\sum _{i=1}^n(y_i-(w^T\ast x_i+b){)}^2$$

in that,$y$ is 预测值，$x$ This is a special expedition to import,$w$ is the amount of weight,$b$ is a biased position，$n$ is the number of training data, $L$ is the loss function.

1.2.1.3 Specific code examples and detailed explanations

Taking Python as an example, the code to implement linear regression is as follows:

import numpy as np

# 训练数据
X = np.array([[1, 2], [2, 3], [3, 4], [4, 5]])
y = np.array([1, 2, 3, 4])

# 初始化权重向量
w = np.random.randn(2, 1)
b = np.zeros(1)

# 学习率
alpha = 0.01

# 迭代次数
iterations = 1000

# 梯度下降
for i in range(iterations):
    # 预测值
    y_pred = np.dot(X, w) + b
    # 损失函数梯度
    grad_w = np.dot(X.T, (y_pred - y))
    grad_b = np.sum(y_pred - y)
    # 更新权重向量
    w = w - alpha * grad_w
    b = b - alpha * grad_b

# 输出预测结果
print("预测结果：", y_pred)

1.2.2 Logistic regression

Logistic regression is a supervised learning algorithm used to predict binary classification target variables. Its core idea is to make the model's prediction of the training data as close as possible to the actual label by finding the optimal weight vector.

1.2.2.1 Core concepts and connections

The core concepts of logistic regression include training data, weight vectors, loss functions, etc.

• Training data: A data set consisting of input features and corresponding labels.
• Weight vector: Parameters used to map input features to target variables.
• Loss function: used to measure the difference between model predictions and actual labels. Commonly used loss functions include Cross-Entropy Loss.

1.2.2.2 Detailed explanation of core algorithm principles and specific operation steps as well as mathematical model formulas

The algorithm principle of logistic regression is to gradually update the weight vector through the gradient descent method to minimize the loss function. The specific steps are as follows:

1. Initialize the weight vector to a random value.
2. For each training data, the predicted value is calculated.
3. Calculate the difference between the predicted value and the actual label.
4. Update the weight vector to minimize the loss function.
5. Repeat steps 2-4 until convergence.

The mathematical model formula is:

$$P(y=1)=\frac{1}{1+{It\; is}^{-(w^T\ast x+b)}}$$

$$L=-\frac{1}{n}\sum _{i=1}^n[y_i\ast \log (P(y_i=1))+(1-{and}_i)\ast \log (1-P(y_i=1))]$$

In that,$P(y=1)$ is the approximate probability of 预测为1，$e$ is cardinal,$n$ is the number of training data, $L$ is the loss function.

1.2.2.3 Specific code examples and detailed explanations

Taking Python as an example, the code to implement logistic regression is as follows:

import numpy as np

# 训练数据
X = np.array([[1, 2], [2, 3], [3, 4], [4, 5]])
y = np.array([[1, 0], [1, 0], [0, 1], [0, 1]])

# 初始化权重向量
w = np.random.randn(2, 1)
b = np.zeros(1)

# 学习率
alpha = 0.01

# 迭代次数
iterations = 1000

# 梯度下降
for i in range(iterations):
    # 预测值
    y_pred = 1 / (1 + np.exp(-(np.dot(X, w) + b)))
    # 损失函数梯度
    grad_w = np.dot(X.T, (y_pred - y))
    grad_b = np.sum(y_pred - y)
    # 更新权重向量
    w = w - alpha * grad_w
    b = b - alpha * grad_b

# 输出预测结果
print("预测结果：", y_pred)

1.2.3 Decision tree

Decision tree is a supervised learning algorithm used to predict continuous target variables. Its core idea is to divide the input features into different subsets by recursively building a tree structure to minimize the uncertainty of the target variable.

1.2.3.1 Core concepts and connections

The core concepts of decision trees include training data, decision trees, information gain, entropy, etc.

• Training data: A data set consisting of input features and corresponding labels.
• Decision tree: A tree-like structure used to divide input features into different subsets.
• Information gain: A measure used to measure the uncertainty reduction of target variables by feature partitioning. Commonly used information gain calculation methods include information entropy (Entropy).
• Entropy: used to measure the uncertainty of the system, ranging from 0 to 1. A small value indicates that the system is relatively certain, and a large value indicates that the system is relatively uncertain.

1.2.3.2 Detailed explanation of core algorithm principles and specific operation steps as well as mathematical model formulas

The algorithm principle of decision tree is to divide the input features into different subsets by recursively building a tree structure to minimize the uncertainty of the target variable. The specific steps are as follows:

1. For each feature, the information gain is calculated.
2. Select the feature with the largest information gain as the split point.
3. For each eigenvalue, a subtree is built recursively.
4. Repeat steps 2-3 until stopping conditions (such as minimum number of samples, maximum depth, etc.) are met.

The mathematical model formula is:

$$Entropy\left(S\right)=-\sum _{i=1}^{n}P(c_i)\ast {\log }_2(P(c_i))$$

$$Gain(S,A)=Entropy\left(S\right)-\sum _{v\in A}\frac{|S_v|}{|S|}\ast Entropy\left(S_v\right)$$

其中，$Entropy\left(S\right)$ 是 data set$S$の熵，$P\left(c_i\right)$ is this$c_i$approximation,$Gain(S,A)$ This special expedition$A$The collection$S$'s information growth.

1.2.3.3 Specific code examples and detailed explanations

Taking Python as an example, the code to implement the decision tree is as follows:

import numpy as np
from sklearn.tree import DecisionTreeRegressor

# 训练数据
X = np.array([[1, 2], [2, 3], [3, 4], [4, 5]])
y = np.array([1, 2, 3, 4])

# 初始化决策树
dt = DecisionTreeRegressor(max_depth=3)

# 训练决策树
dt.fit(X, y)

# 预测结果
y_pred = dt.predict(X)

# 输出预测结果
print("预测结果：", y_pred)

1.2.4 Random Forest

Random forest is a supervised learning algorithm used to predict continuous target variables and consists of multiple decision trees. Its core idea is to randomly select features and training data to achieve a certain degree of independence between multiple decision trees, thereby improving prediction accuracy.

1.2.4.1 Core concepts and connections

The core concepts of random forest include decision trees, randomly selected features, randomly selected training data, etc.

• Decision tree: A tree-like structure used to divide input features into different subsets.
• Randomly select features: When training a decision tree, randomly select a part of features to reduce overfitting.
• Randomly select training data: When training a decision tree, randomly select a part of the training data to increase the diversity of the training data.

1.2.4.2 Detailed explanation of core algorithm principles and specific operation steps as well as mathematical model formulas

The algorithm principle of random forest is to randomly select features and training data to make multiple decision trees have a certain degree of independence, thereby improving prediction accuracy. The specific steps are as follows:

1. For each feature, the information gain is calculated.
2. Select the feature with the largest information gain as the split point.
3. For each eigenvalue, a subtree is built recursively.
4. Repeat steps 2-3 until stopping conditions (such as minimum number of samples, maximum depth, etc.) are met.
5. Use multiple decision trees to make predictions and average the predictions.

The mathematical model formula is:

$${and}_{pred}=\frac{1}{T}\sum _{t=1}^T{f}_t(x)$$

In that,$T$ is the number of decision trees, $f_t(x)$ 是第$The\; predicted\; values\; of\; t$ decision trees.

1.2.4.3 Specific code examples and detailed explanations

Taking Python as an example, the code to implement random forest is as follows:

import numpy as np
from sklearn.ensemble import RandomForestRegressor

# 训练数据
X = np.array([[1, 2], [2, 3], [3, 4], [4, 5]])
y = np.array([1, 2, 3, 4])

# 初始化随机森林
rf = RandomForestRegressor(n_estimators=100, max_depth=3)

# 训练随机森林
rf.fit(X, y)

# 预测结果
y_pred = rf.predict(X)

# 输出预测结果
print("预测结果：", y_pred)

1.2.5 Support vector machine

Support Vector Machine (SVM) is a supervised learning algorithm for classification and regression. Its core idea is to make the model's prediction of the training data as close as possible to the actual label by finding the best support vector.

1.2.5.1 Core concepts and connections

The core concepts of support vector machines include training data, support vectors, kernel functions, loss functions, etc.

• Training data: A data set consisting of input features and corresponding labels.
• Support vector: The closest data point to the class boundary, used to define the class boundary.
• Kernel function: A function used to map input features to a high-dimensional space. Commonly used kernel functions include Radial Basis Function (RBF).
• Loss function: used to measure the difference between model predictions and actual labels. Commonly used loss functions include Squared Loss.

1.2.5.2 Detailed explanation of core algorithm principles and specific operation steps as well as mathematical model formulas

The algorithm principle of the support vector machine is to make the model's prediction of the training data as close as possible to the actual label by finding the best support vector. The specific steps are as follows:

1. For each training data, the predicted value is calculated.
2. Calculate the difference between the predicted value and the actual label.
3. Update support vectors to minimize the loss function.
4. Repeat steps 2-3 until convergence.

The mathematical model formula is:

$$and={In}^T\ast x+b$$

$$L=\frac{1}{2n}\sum _{i=1}^n(y_i-(w^T\ast x_i+b){)}^2$$

in that,$y$ is 预测值，$x$ This is a special expedition to import,$w$ is the amount of weight,$b$ is a biased position，$n$ is the number of training data, $L$ is the loss function.

1.2.5.3 Specific code examples and detailed explanations

Taking Python as an example, the code to implement support vector machine is as follows:

import numpy as np
from sklearn import svm

# 训练数据
X = np.array([[1, 2], [2, 3], [3, 4], [4, 5]])
y = np.array([[1, 0], [1, 0], [0, 1], [0, 1]])

# 初始化支持向量机
svm_clf = svm.SVC(kernel='linear')

# 训练支持向量机
svm_clf.fit(X, y)

# 预测结果
y_pred = svm_clf.predict(X)

# 输出预测结果
print("预测结果：", y_pred)

2 Advantages and Disadvantages of Supervised Learning Algorithms

2.1 Advantages

• High prediction accuracy: Supervised learning algorithms can directly use label information to predict target variables more accurately.
• The model is highly interpretable: Supervised learning algorithms can directly use label information to better understand how the model works.
• Wide range of application scenarios: Supervised learning algorithms can be applied to various types of target variables, such as classification, regression, etc.

2.2 Disadvantages

• Requires label information: Supervised learning algorithms require a large amount of label information, thus limiting their application scope.
• Data quality affects prediction results: Supervised learning algorithms have higher requirements for data quality, so data quality affects the accuracy of prediction results.
• Overfitting problem: Supervised learning algorithms may cause overfitting problems, thereby reducing the accuracy of prediction results.

3 Future trends and challenges

3.1 Future trends

• Big data and deep learning: As the scale of data increases, deep learning technology will become the mainstream of supervised learning algorithms.
• Cross-modal learning: Fusing multiple types of data to improve prediction accuracy.
• Interpretable models: Prioritize model interpretability to improve model interpretability and reliability.

3.2 Challenges

• Data quality issues: How to deal with data quality issues such as missing values ​​and noise to improve the accuracy of prediction results.
• Model interpretability issues: How to interpret complex models into human-understandable forms to improve model interpretability and reliability.
• Algorithm optimization problem: How to optimize the computational efficiency and prediction accuracy of the algorithm to adapt to the processing needs of large-scale data.

4 additional questions

4.1 Frequently Asked Questions and Answers

1. What is the difference between supervised learning and unsupervised learning?

   Supervised learning requires label information, while unsupervised learning does not require label information. Supervised learning can directly use label information to predict the target variable more accurately, while unsupervised learning requires automatically discovering structure from the data, which may lead to less accurate prediction results.

2. What is the difference between support vector machines and random forests?

   Support vector machine is a supervised learning algorithm used for classification and regression. It makes the model's prediction of the training data as close as possible to the actual label by finding the best support vector. Random forest is a supervised learning algorithm used to predict continuous target variables. It consists of multiple decision trees. It makes multiple decision trees have a certain degree of independence by randomly selecting features and training data, thereby improving Forecast accuracy.

3. What is the difference between decision trees and random forests?

   Decision tree is a supervised learning algorithm used to predict continuous target variables. Its core idea is to recursively construct a tree structure and divide the input features into different subsets to minimize the uncertainty of the target variable. . Random forest is a supervised learning algorithm composed of multiple decision trees. It randomly selects features and training data to make multiple decision trees have a certain degree of independence, thereby improving prediction accuracy.

4. What is the difference between linear regression and logistic regression?

   Linear regression is a supervised learning algorithm used to predict continuous target variables. Its core idea is to make the model's prediction of the training data as close as possible to the actual label by finding the optimal weight vector. Logistic regression is a supervised learning algorithm used to predict binary classification target variables. Its core idea is to make the model's prediction of the training data as close as possible to the actual label by finding the optimal weight vector.

5. What is the difference between information gain and entropy?

   Information gain is a measure used to measure the uncertainty reduction of target variables by feature partitioning. Commonly used information gain calculation methods include information entropy. Entropy is used to measure the uncertainty of the system, ranging from 0 to 1. A small value indicates that the system is relatively certain, and a large value indicates that the system is relatively uncertain.

6. What is the difference between kernel function and loss function?

   The kernel function is a function used to map input features to a high-dimensional space. Commonly used kernel functions include radial basis functions. The loss function is a function that measures the difference between the model prediction and the actual label. Commonly used loss functions include squared loss.

7. What are the stopping conditions for a decision tree?

   The stopping conditions of the decision tree include the minimum number of samples, maximum depth, etc. When the stopping conditions are met, the construction process of the decision tree will stop.

8. What is the stopping condition of random forest?

   The stopping conditions of random forest include the minimum number of samples, maximum depth, etc. When the stopping conditions are met, the construction process of the random forest will stop.

9. What is the kernel function of support vector machine?

   The kernel function of a support vector machine is a function used to map input features to a high-dimensional space. Commonly used kernel functions include radial basis functions.

10. What is the loss function of support vector machine?

The loss function of a support vector machine is a function used to measure the difference between the model prediction and the actual label. Commonly used loss functions include square loss.

11. What are the advantages and disadvantages of supervised learning algorithms?

Advantages: high prediction accuracy, strong model interpretability, and wide application scenarios.

Disadvantages: label information is required, data quality affects prediction results, and overfitting problems.

12. What are the future trends of supervised learning algorithms?

Future trends include big data and deep learning, cross-modal learning, interpretive models, etc.

13. What are the challenges of supervised learning algorithms?

Challenges include data quality issues, model interpretability issues, algorithm optimization issues, etc.

14. What are the main techniques of supervised learning algorithms?

Main technologies include linear regression, logistic regression, decision trees, random forests, support vector machines, etc.

15. What are the application scenarios of supervised learning algorithms?

Application scenarios include classification, regression, prediction, etc.

16. What is model evaluation for supervised learning algorithms?

Model evaluation includes training set evaluation, test set evaluation, etc. Commonly used evaluation indicators include accuracy, recall, F1 score, etc.

17. What is feature selection for supervised learning algorithms?

Feature selection is the selection of the most important input features to improve the prediction accuracy of the model. Commonly used feature selection methods include information gain, recursive feature selection, etc.

18. What is cross-validation for supervised learning algorithms?

Cross-validation is a method used to evaluate the performance of a model by dividing the data set into a training set and a test set to evaluate the performance of the model on different subsets of the data. Commonly used cross-validation methods include K-fold cross-validation, Leave-One-Out cross-validation, etc.

19. What is model optimization for supervised learning algorithms?

Model optimization is to improve the prediction accuracy of the model by adjusting the model parameters. Commonly used model optimization methods include gradient descent, stochastic gradient descent, etc.

20. What is the model interpretation of supervised learning algorithms?

Model explanation is to interpret complex models into human-understandable form to improve the interpretability and reliability of the model. Commonly used model interpretation methods include feature importance analysis, model visualization, etc.

21. What are the model choices for supervised learning algorithms?

Model selection is to select the best model to improve the prediction accuracy of the model. Commonly used model selection methods include cross-validation, information Criterion, etc.

22. What are the model parameters for supervised learning algorithms?

Model tuning is to improve the prediction accuracy of the model by adjusting the model parameters. Commonly used model parameter adjustment methods include grid search, random search, etc.

23. What is model synthesis for supervised learning algorithms?

Model synthesis is to fuse the prediction results of multiple models to improve the prediction accuracy of the model. Commonly used model synthesis methods include weighted average, majority voting, etc.

24. What is model fusion for supervised learning algorithms?

Model fusion is to fuse the prediction results of multiple models to improve the prediction accuracy of the model. Commonly used model fusion methods include weighted average, majority voting, etc.

25. What are the model evaluation metrics for supervised learning algorithms?

Model evaluation indicators are indicators used to evaluate model performance. Commonly used model evaluation indicators include accuracy, recall, F1 score, etc.

26. What is model visualization for supervised learning algorithms?

Model visualization is to visualize complex models into a human-understandable form to improve the interpretability and reliability of the model. Commonly used model visualization methods include decision tree visualization, feature importance visualization, etc.

27. What are the model optimization techniques for supervised learning algorithms?

Model optimization techniques are methods used to improve model performance. Commonly used model optimization techniques include regularization, feature engineering, etc.

28. What are the model parameter tuning techniques for supervised learning algorithms?

Model parameter adjustment techniques are methods used to adjust model parameters. Commonly used model parameter adjustment techniques include grid search, random search, etc.